Can someone help me out regarding determine the cut point between two vector lines? (Linear Algebra)

Question

So we have two points: P(1,1,1) and Q(2,3,4) that are found on a line L.

In previous taks we had to find a vector parallell to the line L, which  I found out to be (i+j+k). We also had to find a parametrization of the line which I found to be r (t) = i+2j+3k)t + (i+j+k).

the Line L is however crossing another line given by p(s)=(4i+6j+8k)s and we need to find the point where these lines are crossing each other and here I am totally stuck in what to do.

Someone told me that the clue is to figure out the values of s and t, but when I try to dissect them each, then there is a total of 3 possible values of s and t which makes this much harder than I believe it needs to be.

Could someone give me a hand?

Answer 1

`r(t)` parameterizes points on the first line, and `p(s)` parametrizes points on the second line. You want a point which can be parametrized by either line, so `r(t)=p(s)`

Remember that `i,j,k` are orthogonal, so a point can only be equal to another point if their number of `i` are equal (one equation), their number of `j` are equal (another equation), and their number of `k` are equal as well (third equation). So you get three equations with two variables `t,s`